How Are Geometric Sequences Used to Model Disease Spread or Population Growth?

Geometric sequences model disease spread or population growth when the quantity changes by the same multiplicative factor during each equal time period. If the percentage change remains constant, the population after period $n$ can be represented by $u_n=u_1r^{n-1}$.

The mechanism

A geometric sequence has a constant common ratio $r$, where

$$r=\frac{u_{n+1}}{u_n}.$$

For growth of $p\%$ per period, $r=1+\frac{p}{100}$. For a decrease of $p\%$, .

Suppose there are initially $50$ infected people and the number of infected people increases by $40\%$ each week. Then $u_1=50$ and $r=1.40$, so

$$u_n=50(1.40)^{n-1}.$$

After six weeks,

$$u_6=50(1.40)^5\approx269.$$

The model therefore predicts approximately $269$ infected people in week six.

Similarly, if a population initially contains $1200$ organisms and grows by $8\%$ annually, its population after $t$ years is

$$P(t)=1200(1.08)^t.$$

This uses $t=0$ for the initial population, explaining why the exponent differs from the sequence formula indexed from $u_1$.

The model can also be built from consecutive data values. If a population rises from $800$ to $920$ in one period, then $r=\frac{920}{800}=1.15$, corresponding to growth. The next predicted value is . This demonstrates the modelling cycle: identify a pattern, construct the model, use it, then evaluate its assumptions carefully against real-world data. Predictions within the observed range are generally safer than distant extrapolations.

A common misconception is that real disease or population growth remains geometric indefinitely. The model assumes a constant growth rate and ignores factors such as limited resources, immunity, migration, and control measures, so long-term extrapolation may be unreliable.

IB exam technique

This is SL 1.3, Geometric Sequences and Series, studied at both SL and HL. In an exam, identify $u_1$ and $r$, write the model before using the GDC, retain full precision, and interpret the final value in context. If asked to comment, state whether the constant-ratio assumption is realistic.